Adaptivity with moving grids

68 C. J. Budd, W. Huang and R. D. Russell
the Jacobian J. A measure for the skewness s of the mesh in Ω P is given by
s = λ 1
+ λ 2
= (λ 1 + λ 2 ) 2
− 2= trace(J)
)2
− 2=∆(P − 2, (3.45)
λ 2 λ 1 λ 1 λ 2 det(J) H(P )
where λ 1 and λ 2 are the (real and positive) eigenvalues of J. The skewness
can be estimated in certain cases. One example of this arises in the
scale-invariant meshes for the local singularities blow-up problems studied
in Section 5, in which a sequence of meshes are calculated which, close to
the singularity, take the form P (ξ,t) =Λ(t) ˆP (ξ) for an appropriate scaling
function Λ(t). It is immediate that
∆(P ) 2
H(P ) − 2=∆( ˆP ) 2
H( ˆP ) − 2,
so that the skewness of the rescaled mesh is the same as the original. Hence,
if an initially uniform mesh is used then the mesh close to the singularity
will retain local uniformity.
3.3.3. Solution of the Monge–Ampère equation
The equation (3.41) can be solved either directly (Delzanno et al. 2008) or
by a relaxation method.
The direct method. The Monge–Ampère equation (Evans 1999, Gutiérrez
2001) belongs to the class of fully nonlinear second-order equations and
has two sources of nonlinearity. Firstly, the Hessian H(P ) is nonlinear in
the second derivatives (except in the one-dimensional case). Secondly, the
monitor function in general depends nonlinearly on the first derivatives of
P (either directly or through the solution of the original PDE). For any
suitably smooth positive monitor function the equation has a unique solution,
which is a convex function. Linearization of the equation shows that
it is elliptic in the space of convex functions. The Monge–Ampère equation
arises from prescribing the product of the eigenvalues (the determinant) of
the Jacobian matrix of a gradient mapping. If we prescribe the sum of the
eigenvalues (the trace) then we obtain a standard Poisson equation. For the
Poisson equation, multigrid methods can find the solution of a discretization
on a grid with O(N) unknowns using O(N) operations. Methods for
solving the Monge–Ampère equation aim to obtain the same computational
complexity.
Oliker and Prussner (1988) propose a specially designed discretization
and iterative method which explicitly preserve the convexity of the iterates.
Benamou and Brenier (2000) transform the Monge–Ampère equation
into a time-dependent fluid mechanics problem, which is solved using an
iterative method based on an augmented Lagrangian approach. Dean and
Glowinski (2003, 2004) propose finite element discretizations based on an